We started the year off with a variety of two player games. They range from slight modifications of well known games, like tic-tac-toe on a torus, to stranger games like OutFactor. Finding the winning strategy to each game requires a different kind of mathematical approach. Today we worked on the games in groups and students are invited to continue working on them until our next meeting. At the beginning of our next meeting any student who wants to will present full or partial solutions to any of the games they worked on.

This week we first presented a solution to the game OutFactor that we played last week. We then went on to discuss another game we played last week. The rules are that you start with an n by m grid of dots and players take turns filling in an edge. The game ends when one player is forced to make a cycle. We conjectured that no matter what the players do they will always be able to fill in nm-1 edges and then the next player is forced to make cycle. To prove our conjecture we defined many terms like G-Raff's and tRees and proved a few intuitive but tricky smaller claims.

This week we started with a puzzle from probability: the emperor of ancient China declares that whenever a family gives birth to a girl, they must stop having children. How does the male/female ratio change over time? We saw that the ratio should stay the same, by finding the expected value of the difference between the male and female populations in a couple of different ways. We then continued our investigations into gRaff's, proving that a gRaff with n vertices and fewer than n-1 edges must be disconnected, and helping us get closer to fully understanding the edge coloring game from the first week.

We looked at a different kind of G-raff problem this week known as Ramsey theory. If you two color a compelete graph on six vertices, can you be sure that one of the two colored subgraphs contains a complete graph on two vertices? In general what is the largest complete graph that can be colored so as to not include a complete subgraph of varies types in either of the coloring? This question is suprisingly hard, but with the help of the pigeonhole principle we were able to make some progress.

We had a guest lecture by Austin Tran this week. He gave us a glimpse into the deep and wonderful world of number thoery. To get started we went through the basic of modular arithmatic, including adding, multiplying and division. We then discussed Euler's totient function, Fermat's little theorem, and the idea of psudoprimes, numbers that appear prime under some test but are not actually prime.

Today we discussed many problems relating to geometry on a sphere. To begin we had to decide what a stright line on a sphere should be. Using this we investigated what the sum of the angles of a triangle must be. Suprisingly they can vary from as small as 180 degrees to as large as 540 degrees. We also noticed that there are no parallel lines on a sphere.

We begin by wrapping up some questions about the sphere. It turns out that we can relate the angles of a triangle on a sphere to the area of the triangle, a fact that is not at all true on the plane. This also gave us a proof of the bounds of the sum of the angles that we conjectured last time.
Next we started to delve into voting theory. We spent most of the time discussing different methods one could use to decide a ranking of multiple options with many voters. We then investigated how we could evaluate these voting methods.

We continued our discussion about ways to vote. This time we used a list of criterion to try to evaluate which voting system was "best". We discovered that the system that satisfied the most criterion was Monarchy and that every other system failed some critical test. This led to a breif discussion of Arrow's Impossibility Theorem.

Using a stack of cards instead of dominoes we set out to find how far we could overhang them from the edge of a table by stacking them. It turns out that in theory you can have an infinite amount of overhang! Still it is in interesting question to see how far you can get with a finite number. How many cards do you need to make it one card length off?

Today we talked about how to find all positive integer solutions to the equation x^2+y^2=z^2. We used two methods to do this. The first method worked by reducing modulo 4 and making some number theory arguments. The second worked by noticing that pythagorean triples are the same as rational points on the unit circle. By projecting the circle onto the line we put these points into correspondence with rational points on the line.

We continued our discussion of Fermat's Last Theorem. How many solutions are there to x^n+y^n+z^n when n is not 2? Next we moved on to Pell's equation. How many solutions does "x^2-2y^2 =1" have? By factoring the left hand side and taking powers we were able to find infinitly many solutions.

We started class by considering all the diffeferent types of sufraces you can get by gluing the edges of a rectangular piece of stretchy rubber to each other. In this list are cylindars, mobious bands, spheres, tori, Klein bottles, and something we called a SIG. We talked about ways to compactly write down the gluing data and how some surfaces have multiple representations. We ended by discussing a way to glue two surfaces together by cutting them up and pasting them back together.

We continued our discussion from last week. We were finally able to show that a torus plus a SIG was the same as the sum of three SIG's. To do this we had to be careful about what rules we can use to rewrite a polygonal presentation with. We eneded with a delightful trick in which you cut a mobious band down the center lenghtwise, and then cut one into thirds lenghwise. We were almost able to predict the outcome using polygonal presentations except for some phenomenon that only occur when you embed a surface into space.

We showed that we can classify loops on the plane minus a point by the number of times they go around the missing point. We also discussed loops on a torus, loops on the plane minus two points and we never solved loops on a SIG but I assure you it is very odd.
For a cute riddle, how can you hang up a painting on two nails so that if you remove either nail it falls?

Today we started counting things! We counted the number of ways to choose k objects from a set of n things. We counted the number of ordered partitions of n and we found that it is very difficult to count the number of unordered partitions. However we did make a recursion for the number of unordered partitions of n.
Along the way we showed a lot of bijections, one to one pairings between two sets. An unsolved bijection to think about for next week is the following:
The number unordered partitions with odd parts equals the number of unordered partitions with distinct parts.

We spent today discussing two topics: First, I presented a proof that pi equals four by approximating the parimeter of a circle with a square whose corners have been repeatedly folded in. We discussed why this proof fails. This led us to the idea of a "limit" and we discussed how we might define such a thing.

Back to counting things. This time we focused on bijections and introduced the idea of generating functions to find a closed form equation for fibonacci numbers.
We also had a guest join us and show us some mathematical card tricks!

Today we discussed the riddle of whether or not you can turn a sphere inside out without making any creases. To solve this problem we first talked about turing a circle inside out without any creases and decided it was possible. Along the way we encoutered our old friend, the winding number.

Today we tried to figure out how to multiply points on the plane. First we had to figure out what properties we wanted multiplication to have and then we had several ideas. We tried using matrices to multiply them but found that it was not commuative. We used complex numbers to do the trick and it seemed to work but it was hard to prove nice properites. Finally we switched to polar coordinates and then it was easy to see geometrically that multiplication works well.

We talked about the game of Penney Ante today. This game involves flipping a coin to generate a sequence of heads and tails. We then ask many questions such as what is the expected waiting time of a sequence? Given two sequences which is more likely to come up first?